Theory of Machines — Law of the Machine For a lifting machine, the linear relationship between effort P and load W is called the law of the machine. Which form is correct (m is the slope constant and C is the constant representing friction)?
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AP = mW − C
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BP = m/W + C
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CP = mW + C
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DP = C − mW
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E—
Answer
Correct Answer: P = mW + C
Explanation
Introduction / Context: Real machines require extra effort beyond the ideal proportional term due to friction and other losses. This relationship is captured by the law of the machine, which is a straight line when P is plotted against W.
Given Data / Assumptions:
- P: effort applied; W: load lifted.
- m: proportionality constant (slope); C: constant effort to overcome friction and other losses.
Concept / Approach: Experimentally, P varies linearly with W. The y-intercept at W = 0 equals the constant C (pure friction effort), and slope m represents how much extra effort is needed per unit load.
Step-by-Step Solution:
At W = 0, P must equal friction effort ⇒ P = C (positive). Therefore, the correct linear form is P = mW + C, a straight line with positive intercept and slope.Verification / Alternative check: Plotting test data (P vs W) for hoists or jacks typically yields a line with intercept C > 0, confirming the “+ C” term.
Why Other Options Are Wrong: “P = mW − C” would give negative effort at small loads; “m/W” is not linear in W; “C − mW” implies decreasing effort with increasing load, which is unphysical for lifting machines.
Common Pitfalls: Confusing ideal mechanical advantage (no friction) with the real line including C; misreading axes (P vs W) when interpreting experimental graphs.
Final Answer: P = mW + C.